Theorem phrel 27670

Description: The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
phrel	|- Rel CPreHil OLD

Proof of Theorem phrel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phnv 27669	. . 3 |- ( x e. CPreHil OLD -> x e. NrmCVec )
2	1	ssriv 3607	. 2 |- CPreHil OLD C_ NrmCVec
3		nvrel 27457	. 2 |- Rel NrmCVec
4		relss 5206	. 2 |- ( CPreHil OLD C_ NrmCVec -> ( Rel NrmCVec -> Rel CPreHil OLD ) )
5	2, 3, 4	mp2 9	1 |- Rel CPreHil OLD

Syntax hints:   C_ wss 3574   Rel wrel 5119   NrmCVeccnv 27439   CPreHil OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-oprab 6654  df-nv 27447  df-ph 27668

This theorem is referenced by:  phop  27673